The first term is easily recognized to be asymptotic in distribution to a standard Brownian motion. The reader can
either directly apply Robolledo's martingale central limit theorem, verifying that in the case that integrands and
intensities are bounded all conditions are satisfied, or apply a more direct result, such as theorem (6.2.1) in
Fleming and Harrington [1]}. Under the family of local alternatives, \(\beta ^{\star }_n =b^{\star }/\sqrt{n}\) , then by the comments following expression REF , the second term is easily seen to be
consistent to the drift function listed in expression REF . Therefore the result follows by Slutzky's theorem.
